Capacities on Harmonic Spaces with Adjoint Structure

Abstract

In the classical potential theory, the capacities defined in terms of Green potentials coincide with the capacity defined by Dirichlet integrals; more precisely, for a compact set K in a Greenian domain Ω in ℝ^d,

$$Sup\{ \mu (\Omega )|G\mu \leqq 1,\,Supp\,\mu \subset K\} = \inf \{ \smallint G\mu \,d\mu |G\mu \geqq 1\,on\,K\} = \,\inf \{ D[f]\,|\,f:\,potential\,on\,\Omega \,with\,f \geqq 1\,on\,K\}$$

, where Gμ is the Green potential of μ ≧ 0 on Ω and D[f] is the Dirichlet integral of f.